Peculiar electron dressed states in phosphorene and $\alpha$-T$_3$ lattices

We have recently conducted a theoretical investigation into the response of electron dressed states, i.e., interacting Floquet states arising from the off-resonant coupling of Dirac spin-1 electrons in the $\alpha$-$T_3$ model, to external radiation with various polarizations. In particular, ally, we have analyzed the role played by the parameter $\alpha$ that is a measure of the coupling strength with the additional atom at the center of the honeycomb lattice and which, when varied, continuously gives a different Berry phase. We have found that the electronic properties of the $\alpha$ -$T_3$ model (consisting of a flat band and two cones) could be modified depending on the polarization of the imposed irradiation. It has been shown that a linearly-polarized dressing field induces anisotropy and significantly modifies the Fermi velocities of the valence and conduction subbands differently from that in graphene and depends on the value of $\alpha$, while the flat band remains unaffected. For elliptically-polarized light, energy band gaps are created and the three subbands are altered for all wave vectors and demonstrate a strong dependence on the phase $\phi$. These results lead to possible radiation-generated band structure engineering, as well as experimental and technological realization of such optoelectronic devices.